Preprint

A local limit theorem for random walks on the chambers of \(\tilde A_2\) buildings

James Parkinson and Bruno Schapira

Abstract

In this paper we outline an approach for analysing random walks
on the chambers of buildings. The types of walks that we
consider are those which are well adapted to the structure of
the building: Namely walks with transition probabilities
\(p(c,d)\) depending only on the Weyl distance
\(delta(c,d)\). We carry through the computations for thick
locally finite affine buildings of type
\(\tilde A_2\) to prove a local limit theorem for
these buildings. The technique centres around the representation
theory of the associated Hecke algebra. This representation
theory is particularly well developed for affine Hecke algebras,
with elegant harmonic analysis developed by Opdam. We give an
introductory account of this theory in the second half of this
paper.